Fatigue Crack Paths 2003

develop on maximumshear planes for all metals. For instance, for 304 stainless steel

the critical plane corresponds to maximal tensile strain and usually the plane orientation

depends on the type of loading.

A more consistent approach is obtained by not specifying the critical plane approach

in advance but requiring the maximumof the failure condition to be reached with re

spect to all orientations, thus

(6)

()*c)(,,,maxFFnnnn=γετσn,

where F c represents the critical value reached by the failure condition. The present defi

nition provides the critical plane which is also the extremal plane, so that the critical

condition is not violated on other potential failure planes.

A particular form of Eq. (6) is obtained by applying the strain energy density asso

ciated with the amplitudes of stress and strain components acting on the critical plane,

cf. Glinka et al. [11]

Δ Δ

W

n n τ γ

σ ε

=

⎢⎣⎡

+ Δ Δ n

⎥⎦⎤

*

n

(7)

max)(

c

n

2 2 2 2

.

This parameter represents only a fraction of the strain energy. However, it does not ac

count for the effect of mean stress. An alternative energy condition was proposed by

Chu [10] by combining maximumnormal and shear stresses with the strain amplitudes,

thus

) n n n n ε σ γ τ Δ + Δ max (8)

(

W

=

m a x ) ( 2maxn

.

*c

The cohesive crack model pioneered by Dugdale [12] and Barrenblatt [13] can be re

garded as further extension of the critical plane approach. It is assumed that when the

critical stress or strain condition is reached on the extremal plane, the gradual separation

on this plane is developed, thus generating a damage zone preceding the crack. The

critical stress condition can be assumed in the form

(9)

()[]0,,maxc≤δστσnnFn,

where σc(δ) is the cohesive strength value. The displacement discontinuity on the

critical plane

[]

+ =

(10)

δ

= − = − + , u u u

,

t n δ δ δ

is then associated with the critical stress condition (9), for instance, by the associated

flow rule

∂

∂ ∂

F

F

n

,

,

>

0

c τ λ δ n

c

(11)

= λ σδ

λ

,

n

t

∂

=